To evaluate the series Sn = ∑(n!)^n/(n^n)^2, we need to analyze the behavior of the individual terms and determine if the series converges or diverges.



Let's simplify the terms of the series before proceeding further.

The term (n!)^n can be expressed as (n!)^n = n^n * n! = n^n * n * (n-1) * (n-2) * ... * 3 * 2 * 1.

Similarly, (n^n)^2 can be expressed as (n^n)^2 = n^2n = n^(2n).

Substituting these values into the series, we get:

Sn = ∑(n^n * n!)/(n^(2n))



To determine if the series converges or diverges, we can use the ratio test. Let's apply the ratio test to the series Sn.

Let's define the ratio Rn = [(n+1)^(n+1) * (n+1)!]/[(n^n * n!)] * [(n^(2n))/(n^(2n))].

Simplifying Rn, we get:

Rn = [(n+1)^(n+1) * (n+1)!]/[n^n * n!] * [n^(2n)/(n^(2n))] = [(n+1)^(n+1) * (n+1)!]/[n^n * n! * n^(2n)].



To apply the ratio test, we take the limit as n approaches infinity of the absolute value of Rn.

Let's calculate the limit:

lim(n→∞) |Rn| = lim(n→∞) |[(n+1)^(n+1) * (n+1)!]/[n^n * n! * n^(2n)]|.

Simplifying the expression, we get:

lim(n→∞) |Rn| = lim(n→∞) |[(n+1)^(n+1)]/[n^n * n^(2n)] * [(n+1)!]/[n!]|.

Using the properties of limits and the fact that n! = n * (n-1)!, we can simplify the expression further:

lim(n→∞) |Rn| = lim(n→∞) |(n+1)/n * (n+1)^n * (n+1)/n^(2n)|.



Let's evaluate the limit using the properties of limits.

lim(n→∞) |Rn| = lim(n→∞) |(n+1)/n| * lim(n→∞) |(n+1)^n| * lim(n→∞) |(n+1)/n^(2n)|.

Using the limit definition of e^x, we know that lim(n→∞) (1 + 1/n)^n =